Complex Addition Geometry

Geometric Interpretation of Complex Addition

Complex numbers can be added geometrically using vectors in the complex plane.

Vector Representation

Each complex number \(z = a + bi\) corresponds to a vector from the origin to the point \((a, b)\). The addition of two complex numbers follows the parallelogram law.

The Parallelogram Law

To add \(z_1 = a_1 + b_1i\) and \(z_2 = a_2 + b_2i\):

Geometrically:

1. Draw vectors representing \(z_1\) and \(z_2\) from the origin
2. Complete the parallelogram with these as adjacent sides
3. The diagonal from the origin gives the sum \(z_1 + z_2\)

Geometric Properties

The modulus of the sum satisfies the triangle inequality:

Equality holds when the vectors point in the same direction.

Also, the difference \(z_1 - z_2\) represents the vector from \(z_2\) to \(z_1\):

This geometric interpretation makes complex addition equivalent to vector addition in \(\mathbb{R}^2\).